Leave it to the 2014 New York Giants to lose a game in which Odell Beckham Jr. made one of the greatest plays in NFL history.

The greatness of this prodigy rookie receiver stands in direct contrast to a team that cannot win. No matter what they do, the Giants lose. Even Sunday night, when Beckham, sans cape, went up, up and away for a catch for the ages — truly remarkable, considering he is the youngest Giant of them all.

Beckham in the second quarter put the Giants on his back by leaping to the heavens with a one-handed grab for a 43-yard touchdown that defied gravity and logic, a play that should have galvanized the Giants, inspired them to finally shed their losing ways and beat back the hated Cowboys. It did not, and shame on the Giants for that.

Oh, they fought and battled, but they lost 31-28 when Tony Romo, with all day and night to throw on a Perry Fewell defense that rushed only four players, after nearly eight seconds finally found Dez Bryant in the left corner of the end zone, breaking free of Dominique Rodgers-Cromartie for a 13-yard touchdown catch with 1:01 remaining. The Giants, once leading 21-10, had pulled ahead 28-24 with 3:00 left, completing a 14-play, 93-yard drive with Eli Manning’s 1-yard TD flip to never-used tight end Adrien Robinson. After that, the Giants caved on defense as Romo took the ’Boys 80 yards in only seven plays. That was a delight to many in the crowd of 80,520 wearing Dallas colors and rooting loud and hard for their Cowboys.

“I hope it’s not the greatest catch of all time, I hope I can make more,’’ Beckham said, “but it really means nothing without a win.’’

Without a win is the way the Giants operate. This seems like a formality, but at 3-8, they are officially eliminated from winning the NFC East.

“That hurts bad. … That’s just going to add on to it, you don’t have a chance, you’re not playing well, where do you go from here?’’ wondered Rodgers-Cromartie.

They lost despite brilliance from their rookie receiver. Words might be inadequate to describe what Beckham did, but here goes:

The Giants (3-8) were leading 7-3 and had the ball on the Cowboys’ 43-yard line when Manning, on first down, fired deep down the field to Beckham, who was running along the right sideline, blanketed by cornerback Brandon Carr, who grabbed the rookie as a penalty flag flew for pass interference. Knocked off balance and with Carr grabbing at his left arm, Beckham leaned backwards and reached skyward with his right arm, extending the fingers of his right hand as far as he could as the ball arrived. Beckham seemingly snatched it out of the air with three fingers. The catch was incredible enough, but Beckham was short of the goal line. He managed to just keep his right foot in bounds as he pushed himself into the end zone.

The crowd — those rooting for the Giants, anyway — erupted seeing it live and howled with delight during the various replays.

“That is the best catch I’ve ever seen,’’ running back Rashad Jennings said. “I was on the sideline, I looked, thought he was playing a video game.’’

“Both of them were pretty spectacular,’’ Manning said. “I guess you’ve got to throw it just bad enough where they’ve got to make a great catch.’’

The Giants scored on their first three series and took a 21-10 lead into halftime, but that’s where the music stopped, as they incurred another second-half meltdown. The Cowboys pulled within 21-17 when Cole Beasley took a short pass from Romo, completely turned around cornerback Jayron Hosley and ran 45 yards untouched.

For a brief instant, it looked as if the Giants actually had a response. A crisp drive reached the Dallas 18-yard line, when Manning spotted Preston Parker running free across the middle at the 5-yard line. If Manning hit Parker in stride he would have scored, but Manning fired the ball too high and all Parker could do was leap and get his left hand on it, deflecting it right to safety Barry Church in the end zone. Manning thrust his arms into the air in frustration and after Church took the interception back 45 yards, Manning was knocked to the ground and then slammed his fist to the turf.

“Threw it high, he was a little flatter than I anticipated but no excuse,’’ Manning said. “I got a guy running open, I got to hit him right in the numbers.’’

It wasn’t long before Dez Bryant exploited a blown Giants coverage and Romo hit him for a 31-yard TD to put the Cowboys ahead 24-21. The Giants came back, but it didn’t stick.

“We played hard,’’ coach Tom Coughlin said. “We put ourselves in position to win the game. We didn’t win.’’